ABC is a triangle in which: P - a = 8 cm , P - b = 6 cm , P - c = 4 cm .Find measurement of the largest angle in triangle where 2P = a + b+ cEdit : My answer: 2 P = a + b + c P = a + b + c 2 P − b = a + b + c 2 − b = 6 = a + b + c 2 − 2 b 2 = 6 = a − b + c 2 = 6 = a − b + c = 12 P − a = a + b + c 2 − a = 8 = a + b + c 2 − 2 a 2 = 8 = − a + b + c 2 = 8 = − a + b + c = 16 − a + b + c = 16 a − b + c = 12 2 c = 28 c = 14 P − c = 4 P − 14 = 4 P = 18 P − a = 8 18 − a = 8 a = 10 P − b = 6 18 − b = 6 b = 12 a &lt; b &lt; cThen angle C is largest angle C o s C = 10 2 + 12 2 − 14 2 2 • 10 • 12 = 1 5 Then angle &lt; C = a r c c o s 1 5 = 78.46

Question

ABC is a triangle in which: P - a = 8 cm , P - b = 6 cm , P - c = 4 cm .Find measurement of the largest angle in triangle where 2P = a + b+ cEdit : My answer:  2  P  =  a  +  b  +  c  P  =            a      +      b      +      c        2    P  −  b  =            a      +      b      +      c        2    −  b  =  6  =            a      +      b      +      c        2    −            2      b        2    =  6  =            a      −      b      +      c        2    =  6  =      a    −    b    +    c    =  12  P  −  a  =            a      +      b      +      c        2    −  a  =  8  =            a      +      b      +      c        2    −            2      a        2    =  8  =            −      a      +      b      +      c        2    =  8  =      −    a    +    b    +    c    =  16      −    a    +    b    +    c    =  16      a    −    b    +    c    =  12  2  c  =  28  c  =  14  P  −  c  =  4  P  −  14  =  4  P  =  18  P  −  a  =  8  18  −  a  =  8  a  =  10  P  −  b  =  6  18  −  b  =  6  b  =  12  a  &amp;lt;  b  &amp;lt;  cThen angle C is largest angle  C  o  s  C  =                    10        2            +              12        2            −              14        2                    2      •      10      •      12        =      1    5  Then angle  &amp;lt;  C  =  a  r  c  c  o  s      1    5    =  78.46

stigliy0 · Accepted Answer

c  =  (  P  −  a  )  +  (  P  −  b  )  =  8  +  6  =  14The same procedure we get   a  =  10 and   b  =  12. So  cos  ⁡  γ  =                              10          2                +                  12          2                −                  14          2                            2        ⋅        10        ⋅        12              =            1      5      Thus   γ  =  arccos  ⁡            1      5        =  .  .  .

Jase Howe · Accepted Answer

Well, we know for every triangle:                    (1)                              {                                                    ∠                                  α                  ∘                                +                ∠                                  β                  ∘                                +                ∠                                  γ                  ∘                                =                                  180                  ∘                                                                                                                                                                |                    A                    |                                                        sin                    ⁡                    ∠                    α                                                  =                                                      |                    B                    |                                                        sin                    ⁡                    ∠                    β                                                  =                                                      |                    C                    |                                                        sin                    ⁡                    ∠                    γ                                                                                                                                                                                  |                    A                    |                                    2                                =                                                      |                    B                    |                                    2                                +                                                      |                    C                    |                                    2                                −                2                                  |                  B                  |                                                  |                  C                  |                                cos                ⁡                ∠                α                                                                                                                                                |                    B                    |                                    2                                =                                                      |                    A                    |                                    2                                +                                                      |                    C                    |                                    2                                −                2                                  |                  A                  |                                                  |                  C                  |                                cos                ⁡                ∠                β                                                                                                                                                |                    C                    |                                    2                                =                                                      |                    A                    |                                    2                                +                                                      |                    B                    |                                    2                                −                2                                  |                  A                  |                                                  |                  B                  |                                cos                ⁡                ∠                γ                                                                  Now, we know that       |    A    |    =  P  −  8,       |    B    |    =  P  −  6 and       |    C    |    =  P  −  4                    (2)                              {                                                    ∠                                  α                  ∘                                +                ∠                                  β                  ∘                                +                ∠                                  γ                  ∘                                =                                  180                  ∘                                                                                                                                                                P                    −                    8                                                        sin                    ⁡                    ∠                    α                                                  =                                                      P                    −                    6                                                        sin                    ⁡                    ∠                    β                                                  =                                                      P                    −                    4                                                        sin                    ⁡                    ∠                    γ                                                                                                                                                                                  (                    P                    −                    8                    )                                    2                                =                                                      (                    P                    −                    6                    )                                    2                                +                                                      (                    P                    −                    4                    )                                    2                                −                2                ⋅                                  (                  P                  −                  6                  )                                ⋅                                  (                  P                  −                  4                  )                                ⋅                cos                ⁡                ∠                α                                                                                                                                                (                    P                    −                    6                    )                                    2                                =                                                      (                    P                    −                    8                    )                                    2                                +                                                      (                    P                    −                    4                    )                                    2                                −                2                ⋅                                  (                  P                  −                  8                  )                                ⋅                                  (                  P                  −                  4                  )                                ⋅                cos                ⁡                ∠                β                                                                                                                                                (                    P                    −                    4                    )                                    2                                =                                                      (                    P                    −                    8                    )                                    2                                +                                                      (                    P                    −                    6                    )                                    2                                −                2                ⋅                                  (                  P                  −                  8                  )                                ⋅                                  (                  P                  −                  6                  )                                ⋅                cos                ⁡                ∠                γ                                                                  Which also gives:                    (3)                              {                                                    ∠                                  α                  ∘                                +                ∠                                  β                  ∘                                +                ∠                                  γ                  ∘                                =                                  180                  ∘                                                                                                                                                                P                    −                    8                                                        sin                    ⁡                    ∠                    α                                                  =                                                      P                    −                    6                                                        sin                    ⁡                    ∠                    β                                                  =                                                      P                    −                    4                                                        sin                    ⁡                    ∠                    γ                                                                                                                                                                                  P                    +                    2                                                        P                    −                    4                                                  =                2                ⋅                cos                ⁡                ∠                α                                                                                                          1                +                                  3                                      P                    −                    8                                                  +                                  3                                      4                    −                    P                                                  =                2                ⋅                cos                ⁡                ∠                β                                                                                                          1                +                                  6                                      8                    −                    P                                                  =                2                ⋅                cos                ⁡                ∠                γ                                                                  So, we can write (for example):                    (4)                    1        +                  1          2                −        cos        ⁡        ∠        γ        +                  3                      4            −            P                          =        2        ⋅        cos        ⁡        ∠        β

ABC is a triangle in which: P - a = 8 cm , P - b = 6 cm , P - c = 4 cm . Find measurement of the l

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